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arXiv · 1812.02165

The non-local mean-field equation on an interval

Azahara DelaTorre·Ali Hyder·Luca Martinazzi·Yannick Sire
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Abstract

We consider the fractional mean-field equation on the interval $I=(-1,1)$ $$(-\Delta)^\frac{1}{2} u=\rho\frac{e^{u}}{\int_{I}e^{u}dx},$$ subject to Dirichlet boundary conditions, and prove that existence holds if and only if $\rho <2\pi$. This requires the study of blowing-up sequences of solutions. We provide a series of tools in particular which can be used (and extended) to higher-order mean field equations of non-local type.

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Azahara DelaTorre, Ali Hyder, Luca Martinazzi, Yannick Sire. 2018-12-05. The non-local mean-field equation on an interval. https://arxiv.org/abs/1812.02165

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